MATH STUDENT BOOK. 11th Grade Unit 7

Transcription

1 MATH STUDENT BOOK 11th Grade Unit 7

2 Unit 7 QUADRATIC RELATIONS AND SYSTEMS MATH 1107 QUADRATIC RELATIONS AND SYSTEMS INTRODUCTION 1. DISTANCE FORMULA AND CONIC SECTIONS 5 DISTANCE FORMULA 5 CIRCLE 9 ELLIPSE 1 SELF TEST CONIC SECTIONS AND IDENTIFICATIONS 2 PARABOLA 2 HYPERBOLA 0 IDENTIFICATION OF CONIC SECTIONS 7 SELF TEST 2 1. SYSTEMS, INEQUALITIES, AND APPLICATIONS 6 SYSTEMS OF EQUATIONS 6 INEQUALITIES 50 APPLICATIONS OF CONIC SECTIONS 5 SELF TEST 60 GLOSSARY 65 LIFEPAC Test is located in the center of the booklet. Please remove before starting the unit. 1

3 QUADRATIC RELATIONS AND SYSTEMS Unit 7 Author: Flod Vest, Ed.D. Editors: Richard W. Wheeler, M.A.Ed. Robin Hintze Kreutzberg, M.B.A. Consulting Editor: Robert L. Zenor, M.A., M.S. Revision Editor: Alan Christopherson, M.S. Media Credits: Page 6: DeanDrobot, istock, Thinkstock. 80 N. 2nd Ave. E. Rock Rapids, IA MM b Alpha Omega Publications, Inc. All rights reserved. LIFEPAC is a registered trademark of Alpha Omega Publications, Inc. All trademarks and/or service marks referenced in this material are the propert of their respective owners. Alpha Omega Publications, Inc. makes no claim of ownership to an trademarks and/or service marks other than their own and their affiliates, and makes no claim of affiliation to an companies whose trademarks ma be listed in this material, other than their own. 2

4 Unit 7 QUADRATIC RELATIONS AND SYSTEMS Quadratic Relations and Sstems Introduction Millions of phsical scientists, managerial specialists, social scientists, engineers, and mathematicians have studied and used the quadratic relations known as conic sections. In this LIFEPAC, ou will learn to describe b equation and graph the circle, ellipse, parabola, and hperbola. You will also eamine some of their applications. The stud will begin with the formula for the distance between two points, which is basic to understanding these quadratic relations. Objectives Read these objectives. The objectives tell ou what ou will be able to do when ou have successfull completed this LIFEPAC. When ou have finished this LIFEPAC, ou should be able to: 1. Use the distance formula to calculate the distance between two points in a coordinate plane. 2. Define this circle and ellipse and use the distance formula to derive their equations.. Graph the circle and ellipse and describe essential elements from their equations.. Write the equations of the circle and ellipse from the graphs and from descriptions of the essential elements. 5. Define the parabola and hperbola and use the distance formula to derive their equations. 6. Graph the parabola and hperbola and describe essential elements from their equations. 7. Write the equations of the parabola and hperbola from the graphs and from descriptions of the essential elements. 8. Identif each conic section from its equation. 9. Solve and graph sstems of first- and second-degree equations. 10. Solve and graph second-degree inequalities. 11. Solve application problems involving conic sections. Introduction

5 QUADRATIC RELATIONS AND SYSTEMS Unit 7 Surve the LIFEPAC. Ask ourself some questions about this stud and write our questions here. Introduction

6 Unit 7 QUADRATIC RELATIONS AND SYSTEMS 1. DISTANCE FORMULA AND CONIC SECTIONS The distance formula is needed in deriving the general equations for the four conic sections we shall stud in this LIFEPAC. The use of the distance formula will become clearer to ou as ou stud the circle and ellipse. Section Objectives Review these objectives. When ou have completed this section, ou should be able to: 1. Use the distance formula to calculate the distance between two points in a coordinate plane. 2. Define the circle and ellipse and use the distance formula to derive their equations.. Graph the circle and ellipse and describe essential elements from their equations.. Write the equations of the circle and ellipse from the graphs and from descriptions of the essential elements. DISTANCE FORMULA Suppose that (, ) and P 1 (, -2) are two points in the coordinate plane. You know that some specific distance eists between P 1 and. If we locate an etra point P making a right triangle, we can write b the Pthagorean Theorem: 2 1 (, ) The distance between P 1 and, denoted b P 1 or d, = (distance between P 1 and P ) 2 + (distance between P and ) 2 = [ ()] 2 + [ (-2)] 2 = (7) 2 + (6) 2 = 85, P 1 (, -2) - P (, -2) Thinking along the same lines, we can obtain a more useful and general formula for the distance between P 1 ( 1, 1 ) and ( 2, 2 ). DISTANCE FORMULA P 1 = ( 2 1 ) 2 + ( 2 1 ) 2 2 ( 2, 2 ) P 1 ( 1, 1 ) -2 - P ( 2, 1 ) Section 1 5

7 QUADRATIC RELATIONS AND SYSTEMS Unit 7 See if the results from the distance formula are reasonable b appling the formula to the familiar -5 right triangle in the model. Model 1: (2, )? - -2 (-1, -1) P 1 P (2, -1) - Calculate: P 1 = [2 (-1)] 2 + [ (-1)] 2 = = 25 = 5 Should P 1 be 5? Yes. Model 2: Graph and find the distance between (-1, 2) and (, -5). (-1, 2) 2 d = [ (-1)] 2 + (-5 2) 2 = = 65, (, -5) Model : Graph and find the distance between (-1, 2) and (-1, -5). We could use the distance formula; however, in cases of horizontal or vertical lines, use the absolute value of the difference between the -coordinates or the -coordinates: -5 (-1, 2) d = 2 (-5) = 7 (-1, -5) -5 6 Section 1

8 Unit 7 QUADRATIC RELATIONS AND SYSTEMS when helpful or necessar. 1.1 (1, 2) and (, ) 1.2 (0, 6) and (2, ) 1. (0, 0) and (6, -8) 1. (-, -2) and (-1, -2) 1.5 (0, 0) and (, 0) Section 1 7

9 QUADRATIC RELATIONS AND SYSTEMS Unit (0, 0) and (0, 8) 1.7 (1, 5) and (1, ) Complete these activities. 1.8 From memor, draw a right triangle with P 1 ( 1, 1 ) and ( 2, 2 ) and write the distance formula. 1.9 Use the distance formula to show that the points A 1 (0, ), A 2 (2, 8), and A (-1, 2) are on a straight line. (Hint: Graph the points and show that A A 2 = A 2 A 1 + A A 1.) 8 Section 1

10 Unit 7 QUADRATIC RELATIONS AND SYSTEMS CIRCLE The circle in a coordinate plane can be thought of as a set of points P, with coordinates (, ), that are all a constant radius (distance) from the center. A circle is shown with center at (0, 0) and radius of. We can think of this circle as the set of all points P, with coordinates (, ), that are a distance of units from the center (0, 0). Writing these conditions with the distance formula, we have the following equation of the circle: - P (, ) ( 0) 2 + ( 0) 2 = Squaring both sides, we have an equation of the circle of radius and center (0, 0): - ( 0) 2 + ( 0) 2 = 9 Thinking along these lines about a more general circle with center (h, k) and radius r, we have b the distance formula the equation of the circle that is the set of all points P, with coordinates (, ), that are a distance of r units from the center (h, k): ( h) 2 + ( k) 2 = r This equation ields even more information about the circle, as shown. Circle: ( h) 2 + ( k) 2 = r 2 r P (, ) Center: (h, k) Radius = r Consider the circle ( ) 2 + ( + 6) 2 = 12. B epanding the epressions we have (h, k) = 0, suggesting that an equation of the form A 2 + C 2 + D + E + F = 0 might be the equation of a circle. Section 1 9

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